Discriminating quantum states: the multiple Chernoff distance

We consider the problem of testing multiple quantum hypotheses $\{\rho_1^{\otimes n},\ldots,\rho_r^{\otimes n}\}$, where an arbitrary prior distribution is given and each of the $r$ hypotheses is $n$ copies of a quantum state. It is known that the average error probability $P_e$ decays exponentially to zero, that is, $P_e=\exp\{-\xi n+o(n)\}$. However, this error exponent $\xi$ is generally unknown, except for the case that $r=2$. In this paper, we solve the long-standing open problem of identifying the above error exponent, by proving Nussbaum and Szko\l a's conjecture that $\xi=\min_{i\neq j}C(\rho_i,\rho_j)$. The right-hand side of this equality is called the multiple quantum Chernoff distance, and $C(\rho_i,\rho_j):=\max_{0\leq s\leq 1}\{-\log\operatorname{Tr}\rho_i^s\rho_j^{1-s}\}$ has been previously identified as the optimal error exponent for testing two hypotheses, $\rho_i^{\otimes n}$ versus $\rho_j^{\otimes n}$. The main ingredient of our proof is a new upper bound for the average error probability, for testing an ensemble of finite-dimensional, but otherwise general, quantum states. This upper bound, up to a states-dependent factor, matches the multiple-state generalization of Nussbaum and Szko\l a's lower bound. Specialized to the case $r=2$, we give an alternative proof to the achievability of the binary-hypothesis Chernoff distance, which was originally proved by Audenaert et al.Comment: v2: minor change

Many Hard Examples in Exact Phase Transitions with Application to Generating Hard Satisfiable Instances

This paper first analyzes the resolution complexity of two random CSP models (i.e. Model RB/RD) for which we can establish the existence of phase transitions and identify the threshold points exactly. By encoding CSPs into CNF formulas, it is proved that almost all instances of Model RB/RD have no tree-like resolution proofs of less than exponential size. Thus, we not only introduce new families of CNF formulas hard for resolution, which is a central task of Proof-Complexity theory, but also propose models with both many hard instances and exact phase transitions. Then, the implications of such models are addressed. It is shown both theoretically and experimentally that an application of Model RB/RD might be in the generation of hard satisfiable instances, which is not only of practical importance but also related to some open problems in cryptography such as generating one-way functions. Subsequently, a further theoretical support for the generation method is shown by establishing exponential lower bounds on the complexity of solving random satisfiable and forced satisfiable instances of RB/RD near the threshold. Finally, conclusions are presented, as well as a detailed comparison of Model RB/RD with the Hamiltonian cycle problem and random 3-SAT, which, respectively, exhibit three different kinds of phase transition behavior in NP-complete problems.Comment: 19 pages, corrected mistakes in Theorems 5 and